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scoundrel [369]

3 years ago

12

How do i simplify this

1 answer:

Anna71 [15]3 years ago

6 0

$\bf ~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \left( \cfrac{3^4}{3^6} \right)^2\implies \left( \cfrac{3^{4\cdot 2}}{3^{6\cdot 2}} \right)\implies \cfrac{3^8}{3^{12}}\implies \cfrac{1}{3^{12}\cdot 3^{-8}}\implies \cfrac{1}{3^{12-8}}\implies \cfrac{1}{3^4}\implies \cfrac{1}{81}$

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Step-by-step explanation:

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We know that for a curve defined as:

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where f'(x) = df(x)/dx

Here we know that we have a function that passes through the point (0, 7)

We also know that the slope of the curve at every point is twice the value of the y-coordinate. (remember that the y-coordinate is given by f(x))

Then we have two equations:

f(0) = 7

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From the shape of the equation, we can assume than this is an exponential equation like:

$f(x) = A*e^{k*x}$

Replacing that in the second equation, we get:

$k*A*e^{k*x} = 2*A*e^{k*x}$

From that equation, we can conclude that k = 2

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$f(x) = A*e^{2*x}$

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With this, we can find the value of A.

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